Question

Find the value of $n$ so that vector $4\hat{i}-6\hat{j}+2\hat{k}$ may be parallel to the vector $6\hat{i}+8\hat{j}+n\hat{k}$.

Answer 1

In this problem we need to calculate the value of $n$ whether the given two vectors are parallel to each other. In vector algebra if $\bar{a}$, $\bar{b}$ are parallel to each other then $\bar{a}=\alpha \bar{b}$ where $\alpha$ is a constant. So, we will first assume that both the vectors are parallel and substitute those vectors in the equation $\bar{a}=\alpha \bar{b}$. Now we will compare coefficients of each term individually and calculate the value of $\alpha$. After that we will compare the coefficient of required value and substitute the value of $\alpha$, then we will simplify the equation to get the required result.

Complete step-by-step solution:
Given vectors are $4\hat{i}-6\hat{j}+2\hat{k}$, $6\hat{i}+8\hat{j}+n\hat{k}$.
We know that if $\bar{a}$, $\bar{b}$ are parallel to each other then $\bar{a}=\alpha \bar{b}$ where $\alpha$ is a constant.
Let us assume that the given two vectors are parallel to each other, then we can write
$4\hat{i}-6\hat{j}+2\hat{k}=\alpha \left( 6\hat{i}+8\hat{j}+n\hat{k} \right)$
Comparing the coefficients of $\hat{i}$ in the above equation, then we will get
$4=\alpha \left( 6 \right)$
Simplifying the above equation by using basic mathematical operations, then we will have
$\begin{aligned} & \alpha =\dfrac{4}{6} \\\ & \Rightarrow \alpha =\dfrac{2}{3} \\\ \end{aligned}$
Now comparing the coefficients of $\hat{j}$ in the obtained equation, then we will get
$-6=\alpha \left( 8 \right)$
Simplifying the above equation by using basic mathematical operations, then we will have
$\begin{aligned} & \alpha =-\dfrac{6}{8} \\\ & \Rightarrow \alpha =-\dfrac{3}{4} \\\ \end{aligned}$
In this problem we can observe that the value of $\alpha$ is not the same for the different variables in the given vectors which signifies us that the given vectors are not parallel to each other. So, we can’t find the required solution with this given data.

Note: In this problem we have given that the two vectors are parallel so we have used the equation $\bar{a}=\alpha \bar{b}$. In some cases, they may give that the two vectors are perpendicular to each other. Then we will equate the vector product of the given two vectors to zero mathematically we can write it as $\bar{a}\centerdot \bar{b}=0$. By using the above equation, we can solve the problem.